Potential energy question

I just made the following statement regarding two masses, M and m, in orbit about each other.

"Btw, IIRC [and I may not be... I will check], as the potential energy of the gravitational field of M, wrt to m, is converted to kinetic energy in m, the mass of M decreases according Einstein's mass/energy relationship. As m loses kinetic energy to potential energy the mass M increases. This accounts for the energy storage and exchange.

Am I screwing this up or does memory serve correctly? It makes sense but that is usually a bad sign.

I just made the following statement regarding two masses, M and m, in orbit about each other.

"Btw, IIRC [and I may not be... I will check],

What does "IIRC" mean?

..as the potential energy of the gravitational field of M, wrt to m, is converted to kinetic energy in m, [the mass of M decreases according Einstein's mass/energy relationship.

This is not quite clear to me. What do you mean by "potential energy of the gravitational field of M, wrt to m". "With respec to m"????? The field itself does not have potential energy. There is a mutal potential energy between the two bodies.

Let's assume that you're refering to two bodies in orbit about each other for which the speed of each and the distance between them changes as a function of time. In such a case the kinetic energy and potential energy will change with time and (disgregarding energy carried away in gravitational waves) the total energy of the system will be constant. The proper mass of each body will remain constant.

Am I screwing this up or does memory serve correctly? It makes sense but that is usually a bad sign.

......as the potential energy of the gravitational field of M, wrt to m, is converted to kinetic energy in m, the mass of M decreases according Einstein's mass/energy relationship. As m loses kinetic energy to potential energy the mass M increases. This accounts for the energy storage and exchange.

Am I screwing this up or does memory serve correctly? :

Well your memory probably needs an upgrade, Ivan.
In Newtonian orbital mechanics the masses stay constant. The energy shuttles back & forth from kinetic to potential energy based upon a change in radial distances and velocities.

I think there is a more fundamental interpretation. This is always tough because I used to be relatively certain of what I know and what I don't, but I haven't studied some of this for so long now...scary....

Ivan, your memory is not so bad. Remember the Einstein box? Anyways, it goes like this. Suppose you fire a flash of light with an energy of 13.6ev at a hydrogen atom. This will knock the electron out of orbit and you will be left with a proton and an electron, which collectively have a total mass that is exactly 13.6ev more than a hydrogen atom. This difference is the potential energy residing in the separated particles. If you bring them back together, the reformed hydrogen atom will release this potential energy in another flash of light with an energy of 13.6ev. In the case of gravitational potential energy, the energy used to separate mass m from mass M is stored by both objects. This is, of course, offset by the energy used to move them apart. If you borrow all the energy used from M, its mass will decrease and the mass of m will increase. If you borrow equally from m and M, both will have the same mass after being separated as they did prior to separation. If you steal the energy from say the moon, m and M will both increase in mass while the moon looses mass.

To put it as simply as I can, there is a well defined concept of conservation of energy in general relativity which does however require specific boundary conditions ("asymptotically flat space-times").

However, when a system contains significant gravitational energy, it's usually not possible to localize it in GR. (The exception is a static space-time. Planets orbiting each other aren't static, though.).

Ivan Seeking "As m loses kinetic energy to potential energy the mass M increases. This accounts for the energy storage and exchange."
Pete "The proper mass of each body will remain constant."
Creator "In Newtonian orbital mechanics the masses stay constant"
Chronos "....collectively have a total mass that is exactly 13.6ev more than a hydrogen atom. This difference is the potential energy residing in the separated particles"
Pete, "If you're asking if gravitational energy has a mass related to it then yes, that's quite true"

Confusing isn't it?

In Newton and GR the mass of the bodies stays constant. In Newton energy is kept in a separate account, and KE + PE = const.

In SR the two accounts appear to merge E = mc2
But physicists like to keep them separate anyway and have invariant particle masses, so that energy has now to be kept in the field. Even if this means in the case of a bound system such as an atom then that field has to store negative energy.

In GR particle masses are constant but except in certain static fields energy goes all over the place, it is not locally conserved and the value of a particle's energy cannot be transported from one position to another in the presence of curvature unless there is a time-like killing vector, which generally does not exist.

Therefore the great classical separate principles of the conservation of energy and mass were magnificently united by SR only to be discarded by GR, except in some special contrived situations.

Perhaps GR needs to be modified to restore the situation? I won't bore you again with my solution!

Pete, "If you're asking if gravitational energy has a mass related to it then yes, that's quite true"

If one views a system of two bodies (non-EM radiating massive bodies separated by a distance) as a black box (in the frame of reference of the center of mass of the box), the mass of that box should not change. It should not depend on whether the energies of the bodies inside are potential or kinetic (ie. as gravitational potential becomes kinetic energy as the separation decreases). According to Special Relativity, kinetic energy contributes [itex] KE/c^2[/itex] to the mass of the body. Therefore gravitational potential energy must contibute equally to the mass of the body. Each will contribute to gravitation of the system. So, gravitational potential energy has to comprise part of the mass or inertia of each body.

So it is not a question of a body's mass being converted to kinetic energy or stored as potential energy. Its mass, and therefore its gravitation, is constant. Its mass merely oscillates between rest and relativistic mass (in the frame of reference of the centre of mass of the system).

Throw a ball vertically. It begins with rest mass m and KE E, at the highest point it has 'rest' mass m' and zero KE. Does m = m' as Pete said "The proper mass of each body will remain constant" or not? If it does, as in GR, what has happened to the KE?

The total energy of the projectile is actually given by E = - P0U0 ,
so time dilation g00 comes into play. From the frame of reference of the Centre of Mass of the Earth this compensates for the potential energy component. Energy is conserved because U0 is a killing vector as the metric components are time independent in such a static gravitational field.

However, from the point of view of the projectile the field is not static as in that frame of reference the metric components change with time (they change with position, which changes with time for a moving observer) and there is no killing vector. In GR energy is not conserved and cannot even be easily defined.

Alternatively the rest mass of the projectile might be defined to include PE, as I think you are suggesting Andrew, in which case m = m0exp(Phi) where Phi is the dimensionless Newtonian potential, as 'rest' mass is defined in SCC.

Throw a ball vertically. It begins with rest mass m and KE E, at the highest point it has 'rest' mass m' and zero KE. Does m = m' as Pete said "The proper mass of each body will remain constant" or not? If it does, as in GR, what has happened to the KE?

The mass of a particle is given by the time component of 4-momentum, i.e. [itex]m = \mu dt/d\tau[/itex] where [itex]\mu[/itex] is the proper mass of the particle. Proper mass is an intrinsic property of a particle and is not changed by speed of position in a gravitational field. However proper mass is different than rest mass. This requires explanation - Take as an example a particular kind of gravitational field for which g0k = 0. This is called a "time-orthogonal" gravitational field. In this case

[tex]m = m(v,\Phi) = \frac{\mu}{\sqrt{1 + 2\Phi/c^2 - \beta}}[/tex]

where [itex]\beta = v/c[/itex]. Rest mass is defined as

[tex]m_0 = m(0,\Phi)[/tex]

The proper mass is related to the mass through

[tex]\mu = m(0,0)[/tex]

The total energy of the projectile is actually given by E = - P0U0 ,
so time dilation g00 comes into play. From the frame of reference of the Centre of Mass of the Earth this compensates for the potential energy component. Energy is conserved because U0 is a killing vector as the metric components are time independent in such a static gravitational field.